1. Time series of the day

2. Stat 153 - 11 Sept 2008 D. R. Brillinger Simple descriptive techniques Trend X t =  +  t +  t Filtering y t =  r=-q s a r x t-r Simple moving average s = q, a r = 1/(2q+1) Filters may be in series

3. Differencing y t = x t - x t-1 =  x t "removes" linear trend Seasonal variation model X t = m t + S t +  t S t  S t-s  12 x t = x t - x t-12, t in months

4. Stationary case, autocorrelation estimate at lag k, r k  t=1 N-k (x t - )(x t+k - ) over  t=1 N (x t - )2 autocovariance estimate at lag k, c k  t=1 N-k (x t - )(x t+k - ) / N

7. Departures from assumptions Nonstationarity Trend - OLS Seasonality - trig functions Missing values Outliers